1/4-kleismic is the tuning which leaves octaves and 7s pure and flattens 3 and 5 by (225/224)^(1/4) (225/224 being termed the septimal kleisma.) This is analogous to 1/4-comma meantone, and like 1/4-comma is the minimax tuning.

A tuning using only 5-limit intervals retunes 3 to |15 -7 -1>, 5 to |15 -8 0> and 7 to |55 -30 -2>.

While not very good so far as 7-limit intervals are concerned, a tuning which keeps 2, 3, and 5 pure and uses the sharp value 225/32 in place of 7 is theoretically very important; it expresses the marvel-tempered tone group in terms of its generators very expeditiously.

supporting linear temperaments:

miracle

waage

hanson

wizard

orwell and garibaldi, at a slightly lesser accuracy.

11-limit

name:

marvel

commas:

{225/224, 385/384}

planar wedgie:

<<<1 2 -3 -2 1 -4 -5 12 9 -19|||

mapping:

[<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|]

TOP tuning:

[1200.509, 1901.149, 2785.133, 3370.019, 4149.558]

possible et tunings:

72, 238, 310

minimax tuning:

(26873856/1375)^(1/9) for 3,
(2097152000/1089)^(1/9) for 5

supporting temperaments:

miracle

wizard

hanson

slender

name:

prodigy

commas:

{225/224, 441/440}

planar wedgie:

<<<1 -2 3 -2 6 -6 5 -13 11 -4|||

mapping:

[<1 0 0 -5 -13|, <0 1 0 2 6|, <0 0 1 2 3|]

TOP tuning:

[1200., 1900.058168, 2783.119618]

possible et tunings:

72, 780

supporting temperaments:

miracle

waage

. . . . . . . . .

Example marvel tuning: a 5-limit scale marvel tempered into ~11-limit

... here is a 31-note scale called keenan5, which I decided to reverse engineer. It is not a random collection of notes; it does not even have a complex structure like a ciculating temperament often will. It is straightforward, logical, and yes, mathematical in the way it was evidently constructed.

By running it through Scala's equal temperament fitter, I found it could be fitted, increasingly well, to 31, 41, 72, 125 and 166. This strongly suggests it is 11-limit marvel, which you can discover by putting together the corresponding standard vals.

By fitting it to 166-equal, and then replacing the steps of size 4, 5, and 7 with steps of size 2048/2025, 16875/16384 and 128/125, I produced an algebraically exact 5-limit version. This does exactly what Dave objects to, namely, exhibits the scale as a planar temperament. Approximations to 2,3 and 5 can be used to generate marvel, and hence the 5-limit version tells you what the mapping from 11-limit JI is--the thing I was calling the temperament itself. At this point choosing a particular tuning is a separate question which does not involve what the temperament *is*, namely 11-limit marvel. I say 11-limit because that is what the header information claims for it, but also because of the precise tuning, which turns out to be the 11-limit minimax tuning.

Hence, Dave's scale can be *precisely* defined as the tempering by 11-limit minimax marvel of the following 5-limit scale [arranged by Monzo into tabular format, with monzos, cents, and graphics added]: